Virgil
Posts:
4,479
Registered:
1/6/11
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Re: The Distinguishability argument of the Reals.
Posted:
Jan 4, 2013 4:31 PM
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In article <kc6idc$jc$1@Kil-nws-1.UCIS.Dal.Ca>,
gus gassmann <gus@nospam.com> wrote:
> On 03/01/2013 5:53 PM, Virgil wrote:
> > In article
> > <a60601d5-24a2-4501-a28b-84a7b1e53bac@ci3g2000vbb.googlegroups.com>,
> > WM <mueckenh@rz.fh-augsburg.de> wrote:
> >
> >> On 3 Jan., 14:52, gus gassmann <g...@nospam.com> wrote:
> >>
> >>> Exactly. This is precisely what I wrote. IF you have TWO *DIFFERENT*
> >>> reals r1 and r2, then you can establish this fact in finite time.
> >>> However, if you are given two different descriptions of the *SAME* real,
> >>> you will have problems. How do you find out that NOT exist n... in
> >>> finite time?
> >>
> >> Does that in any respect increase the number of real numbers? And if
> >> not, why do you mention it here?
> >
> > It shows that WM considerably oversimplifies the issue of
> > distinguishing between different reals, or even different names for the
> > same reals.
> >>>
> >>> Moreover, being able to distinguish two reals at a time has nothing at
> >>> all to do with the question of how many there are, or how to distinguish
> >>> more than two. Your (2) uses a _different_ concept of distinguishability.-
> >>
> >> Being able to distinguish a real from all other reals is crucial for
> >> Cantor's argument. "Suppose you have a list of all real numbers ..."
> >> How could you falsify this statement if not by creating a real number
> >> that differs observably and provably from all entries of this list?
> >
> > Actually, all that is needed in the diagonal argument is the ability
> > distinguish one real from another real, one pair of reals at a time.
>
> Exactly. The only reals that matter to Cantor's argument are the
> *countably* many that are assumed to have been written down. There is no
> need (nor indeed an effective way) to distinguish the constructed
> diagonal from *all* the potential numbers that could have been
> constructed that are not on the list, either. Any *one* number not on
> the list shows that the list is incomplete and thus establishes the
> uncountability of the reals.
But try getting WM to see it!
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